The norming sets of \({{\mathcal {L}}}(^2 {\mathbb {R}}^2_{h(w)})\)

Abstract

An element \((x_1, \ldots , x_n)\in E^n\) is called a norming point of \(T\in {{\mathcal {L}}}(^n E)\) if \(\Vert x_1\Vert =\cdots =\Vert x_n\Vert =1\) and \(|T(x_1, \ldots , x_n)|=\Vert T\Vert ,\) where \({{\mathcal {L}}}(^n E)\) denotes the space of all continuous n-linear forms on E. For \(T\in {{\mathcal {L}}}(^n E),\) we define

$$\begin{aligned} \text {Norm}(T)=\{(x_1, \ldots , x_n)\in E^n: (x_1, \ldots , x_n)~\text{ is } \text{ a } \text{ norming } \text{ point } \text{ of }~T\}. \end{aligned}$$

Let \({\mathbb {R}}^2_{h(w)}\) denote the plane with the hexagonal norm with weight \(0<w<1\)

$$\begin{aligned} \Vert (x, y)\Vert _{h(w)}=\max \Big \{|y|, |x|+(1-w)|y|\Big \}. \end{aligned}$$

We classify \(\text {Norm}(T)\) for every \(T\in {{\mathcal {L}}}(^2 {\mathbb {R}}_{h(w)}^2)\).